I am a highly sociable person and would like to travel on Amtrak when there are loads of people having fun being sociable. That's why I won't travel in a sleeping car. I want to go the lounge car and meet people, play cards and other board games, drink and be merry (I hope this is possible in 2018 like it was back in the 1990's).
I just read this article today about a truck on the tracks hit by a long-distance Amtrak train that only had 59 passengers aboard! So few people, who probably were not sociable, would be a disappointment to me.
So is there a resource to find out how many people ride a specific train and when is the ridership the highest?
To be honest, I highly doubt that you'll end up on a train with "loads of people having fun being sociable". For the most part, people keep to themselves, and in my experience, virtually no one wants to sit down and play a board game until the end. Also, they almost never provide games on board, so you would likely have to bring them yourself. I also disagree with your logic that you wouldn't be able to be social in a sleeper. You can still go to the lounge car if you're in a sleeper, so the only real difference is that you won't have as many people near your seat, and in my experience, people try to keep quiet and keep to themselves in the coaches anyway. And remember that a crowded train means more noise, more smells (coach passengers don't have access to a shower), and a more packed lounge car, meaning it could be very hard to find a seat.
There's a difference between a train being crowded and the people on board wanting to socialize, have a few drinks, and play games. I would say that no matter what season or specific day, there will be a sufficient number of people onboard to socialize with, if they are willing to do so, but the fact is that most people don't want to have a few beers with complete strangers, and don't want to sit in the lounge car playing board games or card games. Also note that you're not allowed to consume alcohol in the coaches anyway.
But if you're really dead set on being on a packed train, which I really don't recommend, peak season is usually summertime for most of the western long distance routes. Could you remind us what train you plan on taking?
NOTE: This is all from my personal experience on crowded trains and socializing on Amtrak. I get that I might be in the minority on this one.